Derivatives Lecture Notes

What is a Derivative?

• Derivative: A function that gives the slope at some x-value.
• Example: For the function f(x) = 8, its graph is a straight line at y = 8, having a slope (derivative) of 0.
• Notation: f'(x) represents the derivative of f(x).
• Differentiation Operator: D/dx indicates differentiation with respect to x.

Derivative of Constants

• The derivative of any constant is 0.
• Examples:
  • d/dx[5] = 0
  • d/dx[-7] = 0

Power Rule for Monomials

• Formula: The derivative of x^n is n * x^(n-1).
• Examples:
  • d/dx[x^2] = 2x^(2-1) = 2x
  • d/dx[x^3] = 3x^(3-1) = 3x^2
  • d/dx[x^4] = 4x^(4-1) = 4x^3
  • d/dx[x^5] = 5x^(5-1) = 5x^4*

Constant Multiple Rule

• Formula: The derivative of cf(x) = c * f’(x)
• Example:
  • d/dx[4x^7] = 4 * d/dx[x^7] = 4 * (7x^6) = 28x^6*

Examples of Differentiation

• d/dx[8x^4] = 32x^3
• d/dx[5x^6] = 30x^5
• d/dx[9x^5] = 45x^4
• d/dx[6x^7] = 42x^6

Confirming the Power Rule

• Using the Definition of the Derivative:
  • f’(x) = limit(h→0)[f(x+h) - f(x)]/h
  • Example with f(x) = x^2:
    • f(x+h) = (x+h)^2 = x^2 + 2xh + h^2
    • f’(x) = limit(h→0)[x^2 + 2xh + h^2 - x^2]/h = limit(h→0)[2xh + h^2]/h = limit(h→0)[2x + h] = 2x

Slope at a Point

• Tangent Line: A line that touches the curve at one point.
• Secant Line: A line that touches the curve at two points.
• Example with f(x) = x^2:
  • Slope of Tangent at x=1: 2(1) = 2

Derivative of Polynomial Functions

• General Approach: Differentiate each term separately.
• Example:
  • f(x) = x^3 + 7x^2 - 8x + 6
  • f’(x) = 3x^2 + 14x - 8
• More Practice:
  • f(x) = 4x^5 + 3x^4 + 9x -7
  • f’(x) = 20x^4 + 12x^3 + 9

Finding the Slope at Specific Points

• Example:
  • Given f(x) = 2x^5 + 5x^3 + 3x^2 + 4
  • At x=2, f’(x) = 160 + 60 + 12 = 232

Derivative of Rational Functions

• Rewrite f(x) = 1/x as x^-1.
• Technique: Use the Power Rule.
  • Example:
    • f(x) = 1/x^2 becomes x^-2
    • f’(x) = -2x^-3 = -2/x^3

Derivative of Radical Functions

• Rewrite √x as x^(1/2).
• Example:
  • d/dx[√x] = (1/2)x^(1/2 -1) = (1/2)x^(-1/2) = 1/(2√x)

Complex Examples

• For f(x) = 8/x^4 rewrite as 8x^-4.
• Differentiate: f’(x) = -32x^-5 = -32/x^5

Trigonometric Functions

• Derivative of sin(x) = cos(x)
• Derivative of cos(x) = -sin(x)
• Derivative of tan(x) = sec^2(x)
• Derivative of cot(x) = -csc^2(x)
• Derivative of sec(x) = sec(x)tan(x)
• Derivative of csc(x) = -csc(x)cot(x)

Product Rule

• Formula: (fg)’ = f’g + fg’
• Example:
  • f(x) = x^2, g(x) = sin(x)
  • (x^2sin(x))’ = 2xsin(x) + x^2cos(x)

Quotient Rule

• Formula: (f/g)’ = (gf’ - fg’) / g^2
• Example:
  • if f(x) = 5x + 6, g(x) = 3x - 7
  • (5x + 6 / 3x - 7)’ = [(3x -7)(5) - (5x + 6)(3)] / (3x - 7)^2
  • Simplified result: (Neglect algebra simplification for brevity)